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Semester 1 Independent Synthesis Exam

Attempt before opening the answer key. Budget 3–4 hours, for E01–E11 including review, plus 15–25 minutes for E12 already allocated in the calculus module. Learners may split this into two sittings with the first answers saved. Use a calculator and one page of self-written notes, but no solution key or generated solution. Record any additional assistance and assess that work as assisted. Timing is a planning estimate, not a competence threshold.

Each E item has its own 0–3 gate in the rubric. Explain assumptions, reasoning and limits. No inline answers are included in this paper.

E01: Logic and proof audit​

For a nonempty finite input domain X and output domain Y, negate “every x in X has some valid y in Y.” Give a counterexample relation. Explain why assuming both P and Q and finding no contradiction does not prove P implies Q.

E02: Repair and prove​

A proposed theorem states 2**n > n*n for every integer n>=4. Check the boundary, repair the lower limit, and give a complete induction proof for the repaired strict inequality.

E03: Count with roles​

Twelve distinct people belong to three disjoint groups of four. Select a four-person committee, a chair and a different vice-chair, requiring the two officers to belong to different groups. Count outcomes and explain why none are duplicated.

E04: Graph certificate​

Construct a finite simple undirected graph on eight labelled vertices, all degree three. List its edges or an unambiguous construction and prove the edge count. Explain why the degree-sum condition alone does not construct the graph.

E05: Probability model and bounds​

Insert 20 keys independently and uniformly into 100 buckets. Find the probability that a fixed bucket is empty, expected empty buckets, and the probability that fewer than 15 buckets are empty. Explain why bucket-empty indicators are not independent.

E06: Inference assumptions​

Twenty-five independent normal observations have mean 2.3 and sample standard deviation 0.4. Using t(0.975,24)=2.064, compute a 95% confidence interval for the population mean. Interpret coverage and explain what changes if these are serially correlated benchmark runs.

E07: Exact versus unique solutions​

Consider x+2y+3z=6, 2x+4y+az=b, and x+2y+2z=c. Characterise consistency in terms of a,b,c; give all solutions when consistent. Can any parameter choice yield a unique solution?

E08: Numerical judgment​

For A=diag(1,10**-8), b=(1,10**-8), compare the exact solution to the solution for perturbed right-hand side (1,2*10**-8). Compute the infinity-norm condition number and the relative residual of the perturbed solution against the original system. Explain why a small residual need not mean a small forward error.

E09: Synthesis: triangle estimator​

For the graph with edges 01,12,02,23, let c(e) be the number of common neighbours of the endpoints. Sample k>=1 edges independently, uniformly, with replacement and estimate T by (4/3) times the sample mean of c(e). Compute the exact triangle count, trace(A cubed), expectation and variance of the estimator. Use Chebyshev to give a sufficient k for absolute error at most epsilon with failure probability at most delta, where epsilon>0 and 0<delta<1.

E10: Changed assumptions and handoff​

The implementation for E09 now samples edges without replacement and stops after a time budget; the graph may contain duplicate input edges. Identify which contract, variance formula and result status need review. Give a defensible partial handoff, one discriminating test and one conclusion you cannot yet support.

E11: Integer arithmetic and complete reachability​

A probe starts at slot zero and advances by 14 around 22 slots. Give all canonical times t in [0,22) at which it reaches slot 10. Prove existence and completeness using a gcd certificate and a reduced congruence, state the cycle length, and explain why slot 9 is unreachable. Contrast checking the returned witnesses with proving that no times were omitted. This item reuses the integer module defense allocation; do not count it as a second project.

E12 — Calculus transfer and numerical judgment​

Normalize f(t)=c t on [0,3], zero outside, and derive its full CDF and mean. Find P(T>1). Explain why f(3) is not a point probability. For numerical integration of t² on [0,3], derive a trapezoid count sufficient for absolute error at most 0.01, stating smoothness assumptions. Reuse your calculus appendix for the numerical comparison; no duplicate experiment is required. Budget 15–25 minutes within the calculus assessment allowance.